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\begin{frame}
\frametitle{Continuity: Examples}

\begin{block}{}
A function $f$ is \emph{continuous form the right} at a number $a$ if
\begin{talign}
\lim_{x\to a^+} f(x) = f(a)
\end{talign}
\end{block}
\pause
\begin{block}{}
A function $f$ is \emph{continuous form the left} at a number $a$ if
\begin{talign}
\lim_{x\to a^-} f(x) = f(a)
\end{talign}
\end{block}
\pause\smallskip

\begin{minipage}{.58\textwidth}
\begin{exampleblock}{}
Where is $\lfloor x \rfloor$ (dis)continuous?
\begin{talign}
\lfloor x \rfloor = \text{\;the largest integer $\le x$\;'}
\end{talign}\vspace{-2ex}
\begin{itemize}
\item<4-> discontinuous at all integers
\item<5-> left-discontinuous at all integers\\
$\lim_{x\to n^-} \lfloor x \rfloor = n-1 \ne n = f(n)$
$\lim_{x\to n^+} \lfloor x \rfloor = n = f(n)$
\end{itemize}
\end{exampleblock}
\begin{minipage}{.39\textwidth}
\scalebox{.75}{
\begin{tikzpicture}[default]
\diagram{-.5}{4}{-.5}{3.2}{1}
\diagramannotatez
\diagramannotatex{1,2,3}
\diagramannotatey{1,2}
\foreach \x in {0,1,2,3} {
\draw[cred,ultra thick] (\x,\x) -- ++(1,0);
\node[include={cred}] at (\x,\x) {};
\node[exclude={cred}] at (\x+1,\x) {};
}
\end{tikzpicture}
}
\end{minipage}
\end{frame}
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